A family of 'etale coverings of the affine line, by Kirti Joshi
This paper has appeared in J. Number Theory 59 (1996), no. 2, 414-418,
and so the dvi version has been removed. In this note we show that one can use Drinfel'd modular curves
to construct a family of etale coverings of the affine line. This
leads to construction of a profinite quotient of the algebraic
fundamental group of the affine line.
Our result is proved by using the moduli of Drinfel'd $A$-modules of
rank two over $C$ with $I$-level structure (where $C$ is a suitably
large field ); these Drinfel'd modular curves give rise to a tower of
galois coverings of the affine line ramified only at one point (on the
linei ), and the ramification is tame and independent of $I$. The tame
ramification can be removed in the entire tower by invoking a suitable
variant of Abhyankar's lemma; this variant of Abhyankar's lemma also
calculates the Galois group in our context.
A similar construction leads to a tower of coverings of the
affine space of any dimension.
This note will appear in Journal of Number Theory.
Kirti Joshi <email@example.com>